Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating socio-technical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value – and at the same time fundamentally limited – in their ability to characterize system performance. We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research. Acknowledgments The authors thank Lee Schwarz, Wallace Hopp and the editorial board of M&SOM for initiating this project, as well as the referees for their valuable comments. Thanks are also due to L. Brown, A. Sakov, H. Shen, S. Zeltyn and L. Zhao for their approval of importing pieces of [36, 112].

Deterministic fluid models are developed to provide simple first-order performance descrip-tions for multi-server queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multi-server queues with a large number of servers and non-exponential service-time and time-to-abandon distributions. The first fluid model serves as an approxi-mation for the G/GI/s + GI queueing model, which has a general stationary arrival process with arrival rate λ, independent and identically distributed (IID) service times with a general distribution, s servers and IID abandon times with a general distribution. The fluid model is useful in the overloaded regime, where λ> s, which is often realistic because only a small amount of abandonment can keep the system stable. Numerical experiments, using simulation for M/GI/s + GI models and exact numerical algorithms for M/M/s + M models, show that the fluid model provides useful approximations for steady-state performance measures when the system is heavily loaded. The fluid model accurately shows that steady-state performance depends strongly upon the time-to-abandon distribution beyond its mean, but not upon the service-time distribution beyond its mean. The second fluid model is a discrete-time fluid model, which serves as an approximation for the Gt(n)/GI/s + GI queueing model, having a state-dependent and time-dependent arrival process. The discrete-time framework is exploited to prove that properly scaled queueing processes in the queueing model converge to fluid functions as s → ∞. The discrete-time framework is also convenient for calculating the time-dependent fluid performance descriptions. Subject classifications: Queues, approximations: multi-server queues with abandonment. Queues, multichannel: approximation of non-Markovian multichannel queues with customer abandon-ment.

Motivated by the desire to understand the performance of service-oriented call centers, which often provide low-to-moderate quality of service, this paper investigates the efficiency-driven (ED) limiting regime for many-server queues with abandonments. The starting point is the realization that, in the presence of substantial customer abandonment, call-center service-level agreements (SLA’s) can be met in the ED regime, where the arrival rate exceeds the maximum possible service rate. Mathematically, the ED regime is defined by letting the arrival rate and the number of servers increase together so that the probability of abandonment approaches a positive limit. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ> 1 (so that the arrival rate exceeds the maximum possible service rate). Even though the probability of delay necessarily approaches 1 in the ED regime, the ED regime can be realistic because, due to the abandonments, the delays need not be excessively large. This paper establishes ED many-server heavy-traffic limits and develops associated ap-proximations for performance measures in the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces and exponential abandon times (the final +M). In the ED regime, essentially the same limiting behavior occurs when the abandonment rate α approaches 0 as when the number of servers s approaches ∞; in-deed, it suffices to assume that s/α → ∞. The ED approximations are shown to be useful by comparing them to exact numerical results for the M/M/s/r + M model obtained using an algorithm developed in Whitt (2003), which exploits numerical transform inversion.

An algorithm is developed to rapidly compute approximations for all the standard steady-state performance measures in the basic call-center queueing model M/GI/s/r+GI, which has a Poisson arrival process, IID service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies of call centers indicate that the service-time and abandon-time distributions often are not nearly exponential, so that it is important to go beyond the Markovian M/M/s/r + M special case, but the general service-time and abandon-time distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an ap-propriate Markovian M/M/s/r + M(n) queueing model, where M(n) denotes state-dependent abandonment rates. After making an additional approximation, steady-state waiting-time dis-tributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the aban-donment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.

Consider a sequence of stationary GI/D/N queues indexed by N with servers' utilization 1 #/ # N , # > 0. For such queues we show that the scaled waiting times NWN converge to the (finite) supremum of a Gaussian random walk with drift -#.

We establish heavy-traffic stochastic-process limits for queue-length, waiting-time and over-flow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. In order to capture the impact of the service-time distribution beyond its mean within a Markovian framework, we consider a special class of service-time distribu-tions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These service-time distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981), Puhalskii and Reiman (2000) and Garnett, Mandelbaum and Reiman (2000), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities ρn so that √ n(1 − ρn) → β for − ∞ < β < ∞. To treat finite waiting rooms, we let mn / √ n → κ for 0 < κ ≤ ∞. With the special H ∗ 2 service-time distribution, the limit processes are one-dimensional Markov processes, behaving like diffusion processes with different drift and diffusion functions in two different re-gions, above and below zero. We also establish a limit for the G/M/n/m + M model, having exponential customer abandonments.

informs ® doi 10.1287/opre.1040.0136 © 2004 INFORMS We develop a diffusion approximation for the queue-length stochastic process in the G/GI/n/m queueing model (having a general arrival process, independent and identically distributed service times with a general distribution, n servers, and m extra waiting spaces). We use the steady-state distribution of that diffusion process to obtain approximations for steady-state performance measures of the queueing model, focusing especially upon the steady-state delay probability. The approximations are based on heavy-traffic limits in which n tends to infinity as the traffic intensity increases. Thus, the approximations are intended for large n. For the GI/M/n/ � special case, Halfin and Whitt (1981) showed that scaled versions of the queue-length process converge to a diffusion process when the traffic intensity �n approaches 1 with �1 − �n � √ n → � for 0 <�<�. A companion paper, Whitt (2005), extends that limit to a special class of G/GI/n/mn models in which the number of waiting places depends on n and the service-time distribution is a mixture of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. Finite waiting rooms are treated by incorporating the additional limit mn / √ n → � for 0 <� � �. The approximation for the more general G/GI/n/m model developed here is consistent

This paper proposes simple methods for staffing a single-class call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multi-server queue with customer abandonment, allowing non-exponential service-time and time-to-abandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customer-abandonment and customer-waiting costs, but attention is also given to the standard deviation of the return. The approach is to approximate the performance and the net return, conditional on the random model-parameter vector, and then uncondition to get the desired results. Two recently-developed approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birth-and-death model, having state-dependent death rates. Key words: model-parameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models

In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.

Abstract: In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We